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Risk Adjusted X-bar Chart. Based on Work of Eric Eisenstein and Charles Bethea, The use of patient mix-adjusted control charts to compare in hospital costs of care Health Care Management Science, 2 (1999), 193-198. Farrokh Alemi, Ph.D. Why Chart Data?. To discipline intuitions

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risk adjusted x bar chart

Risk Adjusted X-bar Chart

Based on Work of Eric Eisenstein and Charles Bethea, The use of patient mix-adjusted control charts to compare in hospital costs of care Health Care Management Science, 2 (1999), 193-198

Farrokh Alemi, Ph.D.

why chart data
Why Chart Data?
  • To discipline intuitions
  • To communicate data in vivid graphical ways

Decision makers often attribute positive outcomes to their own skills and negative outcomes to others, while in reality both could be due to chance

data needed
Data Needed
  • Data collected over time
  • Risk (expected outcomes) for each patient
  • Outcomes for each patient measured as a continuous variable

The purpose is to improve not to get so lost in measurement to loose sight of improvement.

what is risk
What Is Risk?
  • A patient’s condition or characteristics that affects the expected outcomes for the patient
    • A severity index used to predict patient outcomes
    • Clinicians’ consensus regarding expected outcomes
    • Patient’s self rating of expected outcomes
example observed expected costs

Number of cases

Expected cost predicted from severity of the patient’s illness or based on experts’ consensus.

Example: Observed & Expected Costs
elements of a control chart
Elements of a Control Chart
  • X axis shows time
  • Y axis shows average cost (or dependent variable of interest)
  • Observed rates are plotted against time sequence
  • Upper or lower control limit are drawn so that points 95% or 99% of data should fall within these limits
steps in creating x bar chart
Steps in Creating X-bar Chart
  • Check assumptions
  • Calculate average costs and plot them
  • Calculate average expected costs
  • Calculate standard deviation of difference of observed and expected cost
  • Calculate control limits and plot them
  • Interpret findings
  • Distribute chart
step one check assumptions
Step One: Check Assumptions
  • We are examining continuous variables measured on a ratio or interval scale, e.g. cost, satisfaction ratings, blood pressure, etc.
  • Observations are independent from each other. This assumption is violated if current observations can accurately predict future values.
  • More than 5 observations for each time period.
check normal distribution
Check Normal Distribution
  • Histogram the observed costs
  • Eyeball test: Is the shape bell shaped curve with most data in the middle and little data in both tails
  • For more precise verification of assumption you can do statistical tests of Normal distribution
check equality of variance
Check Equality of Variance
  • Eyeball test: Accept the assumption if ranges are within the same ball park (No range several multiple of the other ranges)
  • For more precise test of the assumption you can do statistical test of equality of variances
step 2 calculate average cost
Step 2: Calculate Average Cost

Cij = Cost of case “i” in time period “j”

nj = Number of cases in time period “j”

Cj = Average cost for time period “j” =  i=1, … njCij / nj

Plot of average costs

plot of the observed rates
Plot of the Observed Rates
  • A graph helps us see possible relationships. Maybe August was a low cost month.
  • Wait, until you see control limits of what could have been expected.
step 3 average expected costs
Step 3: Average Expected Costs

Eij = Expected cost of case ‘i’ in time period “j”

Ej = Expected cost for time period “j”

Ej = (i=1,…,nj Eij ) / nj

plot expected costs
Plot Expected Costs

Plotting expected cost helps interpret the observed costs but does not settle the question of whether differences are due to chance.

step 4 standard deviation of differences
Step 4: Standard Deviation of Differences

Dij = Difference of observed and expected cost of case ‘i’ in time period “j”

D = Average difference of observed and expected cost for all cases in all time periods

S = Standard deviation of differences

S = [i=1,…,nj j = 1, …m (Dij-D)2 / (n-1)]0.5

Sj = Standard deviation of differences for time period “j”

Sj = S/(nj)0.5

See sample calculation

standard deviation of difference
Standard Deviation of Difference

A. Calculate differences for each case

B. Calculate standard deviation of differences

C. Calculate standard deviation of differences in each time period

step 5 calculate limits
Step 5: Calculate Limits

UCLj = Upper control limit for time period “j”

LCLj = Lower control limit for time period “j”

UCLj = Ej + t * Sj

LCLj = Ej - t * Sj

t = Constant based on t-student distribution

control limits for first period
Control Limits for First Period
  • UCL1 = 335.81 + 3.2 * 20.8
  • LCL1 = 335.81 – 3.2 * 20.8


Negative limits are set to zero as negative costs are not possible

step 6 interpret findings
Step 6: Interpret Findings
  • Two points are outside limits.
  • In these months, costs were different from what could be expected from patients’ severity of illness.
step 7 distribute control chart
Step 7: Distribute Control Chart
  • Include in the information:
    • How was severity measured and expected costs anticipated?
    • Why are assumptions met?
    • What does the control chart look like?
    • What is the interpretation of the findings?
summary of steps
Summary of Steps
  • Check assumptions
  • Calculate and plot observed cost
  • Calculate expected cost
  • Calculate standard deviation of differences
  • Calculate and plot control limits
  • Interpret findings
  • Distribute control chart